Abstract
In this article, we discuss the long-time dynamical behavior of the stochastic non-autonomous nonclassical diffusion equations with linear memory and additive white noise in the weak topological space . By decomposition method of the solution, we give the necessary condition of asymptotic compactness of the solutions, and then prove the existence of random attractor, while the time-dependent forcing term only satisfies an integral condition.
Highlights
We investigate the asymptotic behavior of solutions to the following stochastic nonclassical diffusion equations driven by additive noise and linear memory:
S (t ) t≥0 be the solution operator of equation (3.3), and the conditions of the lemma 3.6 hold, the random dynamical system Υ has a unique random attractor in 1
Summary
We investigate the asymptotic behavior of solutions to the following stochastic nonclassical diffusion equations driven by additive noise and linear memory: uut= (−x,∆t )ut. In [8] the authors considered the nonclassical diffusion equation with hereditary memory on a 3D bounded domains. In [10] the authors proved the existence and the regularity of time-dependent global attractors for a class of nonclassical reaction-diffusion equations when the forcing term g ( x) ∈ H −1 (Ω). The researchers in [12] obtained the Pullback attractors for the nonclassical diffusion equations with the variable delay on a bounded domain, where the nonlinearity is at most two orders growth. In [14] Ma studied the existence of global attractors for nonclassical diffusion equations with the arbitrary order polynomial growth conditions. The case of μ ≠ 0 with additive noise on a bounded domain, Cheng used the decomposition method of the solution operator to consider the stochastic nonclassical diffusion equation with fading memory. In Section three, firstly, we define a continuous random dynamical system to proving the existence and uniqueness of the solution, prove the existence of a closed random absorbing set and establish the asymptotic compactness of the random dynamical system prove the existence of -random attractor
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